Let's simplify each of these expressions by combining like terms:

1. Expression: $(3 - 2x + 4y) - (6 + 3x - 5y)$

   Solution:

   = (3 - 6) + (-2x - 3x) + (4y + 5y) = -3 - 5x + 9y$$ **Simplified Expression:** $$-3 - 5x + 9y$$

2. Expression: $(4x + 3y - 5z) + (5x - 3y + 6z)$

   Solution:

   = (4x + 5x) + (3y - 3y) + (-5z + 6z) = 9x + z$$ **Simplified Expression:** $$9x + z$$

3. Expression: $(x^2 - y^2 + xz) - (y^2 - x^2 + xz) - (x^2 - y^2)$

   Solution:

   = (x^2 + x^2 - x^2) + (-y^2 - y^2 + y^2) + (xz - xz) = x^2 - y^2$$ **Simplified Expression:** $$x^2 - y^2$$

4. Expression: $(2p^2 - 3q^2 - 4r^2) + 3p^2 - 5q^2 - 5r^2$

   Solution:

   = (2p^2 + 3p^2) + (-3q^2 - 5q^2) + (-4r^2 - 5r^2) = 5p^2 - 8q^2 - 9r^2$$ **Simplified Expression:** $$5p^2 - 8q^2 - 9r^2$$

5. Expression: $[4a^2b - 3ab^2 - (3a + 4b)] - (5a^2b + 2ab^2 - 2a - 3b)$

   Solution:

   = (4a^2b - 5a^2b) + (-3ab^2 - 2ab^2) + (-3a + 2a) + (-4b + 3b) = -a^2b - 5ab^2 - a - b$$ **Simplified Expression:** $$-a^2b - 5ab^2 - a - b$$

Would you like more details on any of these steps or have any questions?

Related Questions:

1. How do you identify like terms in an algebraic expression?
2. What is the significance of the order of operations when combining like terms?
3. Can you explain why certain terms cancel each other out during simplification?
4. How does the distributive property apply when simplifying expressions?
5. What are some common mistakes to avoid when simplifying algebraic expressions?

Tip:

Always double-check the signs (positive or negative) when combining like terms to avoid errors in simplification.